Integrity is doing what you say, and saying what you think. Your thoughts, your words, your actions all match up. I preached good 3 point sermons, tying in that the students needed to be careful that they kept the rules (I was the Dean of Students, so there were reasons for talking about this). :-)
Speaking about integrity often, I had to work hard to guard mine. I didn't want to do anything that might make people think that I was dishonest or cheating someone, or having something come out of my mouth that contradicted my actions. I still work not that to this very day.
Job had his integrity questioned by his friends, because in their world view, bad things only happened to bad people, so since so many bad things were happening to job, he must be a bad dude. But he refused to agree with them. In chapter 27 he says:
Job had spent a lifetime maintaining his integrity. When he lost everything, it was about all he had to hold on to, but he had that.
I hope that none of us find us in the position Job was in, but if we do, it's my prayer that we will all have integrity to which we can "hold fast."
(You saw that my novel is about to be published, right? You can preorder by clicking here).